A NUMERICAL STUDY OF THE HOMOGENEOUS ELLIPTIC EQUATION WITH FRACTIONAL BOUNDARY CONDITIONS

作者：Lazarov Raytcho; Vabishchevich Petr

来源：Fractional Calculus and Applied Analysis, 2017, 20(2): 337-351.

DOI：10.1515/fca-2017-0018

• fractional partial differential equations
• fractional boundary conditions
• numerical methods for fractional powers of elliptic operators
• ultra-parabolic equations

We consider the homogeneous equation Au = 0, where A is a symmetric and coercive elliptic operator in H1(Omega) with Omega bounded domain in Rd. The boundary conditions involve fractional power alpha, 0 < alpha < 1, of the Steklov spectral operator arising in Dirichlet to Neumann map. For such problems we discuss two different numerical methods: (1) a computational algorithm based on an approximation of the integral representation of the fractional power of the operator and (2) numerical technique involving an auxiliary Cauchy problem for an ultra-parabolic equation and its subsequent approximation by a time stepping technique. For both methods we present numerical experiment for a model two-dimensional problem that demonstrate the accuracy, efficiency, and stability of the algorithms.

• 出版日期2017-4